Problem: Given $ m \angle ABC = 8x - 21$, and $ m \angle CBD = 2x + 111$, find $m\angle CBD$. $B$ $A$ $D$ $C$
From the diagram, we see that together ${\angle ABC}$ and ${\angle CBD}$ form ${\angle ABD}$ , so $ {m\angle ABC} + {m\angle CBD} = {m\angle ABD}$ Since $\angle ABD$ is a straight angle, we know ${m\angle ABD = 180}$ Substitute in the expressions that were given for each measure: $ {8x - 21} + {2x + 111} = {180}$ Combine like terms: $ 10x + 90 = 180$ Subtract $90$ from both sides: $ 10x = 90$ Divide both sides by $10$ to find $x$ $ x = 9$ Substitute $9$ for $x$ in the expression that was given for $m\angle CBD$ $ m\angle CBD = 2({9}) + 111$ Simplify: $ {m\angle CBD = 18 + 111}$ So ${m\angle CBD = 129}$.